## Abstract

We develop a tractable method to estimate multiple prior models of decision-making under ambiguity. In a representative sample of the U.S. population, we measure ambiguity attitudes in the gain and loss domains. We find that ambiguity aversion is common for uncertain events of moderate to high likelihood involving gains, but ambiguity seeking prevails for low likelihoods and for losses. We show that choices made under ambiguity in the gain domain are best explained by the α-MaxMin model, with one parameter measuring ambiguity aversion (ambiguity preferences) and a second parameter quantifying the perceived degree of ambiguity (perceptions about ambiguity). The ambiguity aversion parameter *α* is constant and prior probability sets are asymmetric for low and high likelihood events. The data reject several other models, such as MaxMin and MaxMax, as well as symmetric probability intervals. Ambiguity aversion and the perceived degree of ambiguity are both higher for men and for the college-educated. Ambiguity aversion (but not perceived ambiguity) is also positively related to risk aversion. In the loss domain, we find evidence of reflection, implying that ambiguity aversion for gains tends to reverse into ambiguity seeking for losses. Our model’s estimates for preferences and perceptions about ambiguity can be used to analyze the economic and financial implications of such preferences.

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## Notes

We chose to implement our ambiguity question involving losses without real incentives to avoid house money effects (Thaler and Johnson 1990). The house money effect refers to empirical evidence that people’s risk taking can depend on prior gains and losses. In our setting, if we had given the respondent an initial endowment to implement real losses, the presence of an endowment could influence people’s subsequent decisions (e.g., more risk taking after a windfall). Exposing respondents to real losses without giving an initial endowment raises ethical issues.

A related paper by Attanasi et al. (2014) tests the behavioral predictions of the smooth model using several decision tasks with varying levels of ambiguity (perceived ambiguity).

The sophisticated approach of Hey et al. (2010) involves joint estimation of risk and ambiguity preferences, as well as beliefs; implementation used 135 choice problems per subject (students). As our objective is to measure ambiguity preferences and perceptions in a survey of the general population, we use a relatively simple elicitation method involving only 17 choices per subject that does not require joint estimation of risk preferences.

Similarly, Conte and Hey (2013) evaluate the predictive ability of several multiple prior models based on subjects’ choices between two compound lotteries with known probabilities (i.e., two-stage lotteries).

For more information about how the ALP recruits respondents see the American Life Panel website at https://mmicdata.rand.org/alp/. One advantage of the ALP is that respondents who lack Internet access are provided with either a laptop and Internet access, or a so-called WebTV that allows them to use their television to participate in the panel.

Dimmock et al. (2015a) describe the fielding of the ALP module in more detail.

The survey module uses “box” instead of “urn,” as the word “urn” might be unfamiliar to some.

In August 2013, we fielded an additional survey (

*N*= 500) identical to the original except one: it offered some respondents a choice for the winning ball color. Specifically, a randomly selected half of the sample was allowed to select the winning color (purple or orange), while the other half could not. Fewer than one percent of the respondents in the group allowed to change the color did so, and the mean matching probabilities of the ‘color choice’ and ‘no color choice’ subgroups were not significantly different. Results are available upon request.Similarly, Baillon and Bleichrodt (2015) find no significant differences between ambiguity attitudes in the loss domain measured with real incentives and with hypothetical losses.

Butler et al. (2014) report that 52% of 1686 Italian bank customers are ambiguity averse (without real incentives), while in Akay et al. (2012) 57% of 92 Ethiopian farmers are ambiguity averse. In a sample of 666 subjects from the Dutch population, Dimmock et al. (2015b) find that 68% were ambiguity averse, 10% neutral, and 22% seeking, suggesting that ambiguity aversion is more common among the Dutch.

A signed rank test for the median gives similar results:

*p*-value < 0.01 for no reference dependence (*median AA*^{50}*= median AA*^{−50});*p*-value = 0.13 for reflection (*median AA*^{50}= −*median AA*^{−50}).*E*^{c}contains all states*s*except those contained in event*E*:*E*∪*E*^{c}*= S*and*E ∩ E*^{c}= ∅.Note that

*P*(*E*) = 0 for*E*∈*N*, where*N*denotes a set of all null events*E*∈*N*with*π*(*E*) = 0.Given sufficient experimental time one could elicit a subjective measure

*π*for uncertain events from revealed preferences at the individual level as in Abdellaoui et al. (2011).The measures of ambiguity aversion and a-insensitivity introduced by Abdellaoui et al. (2011), named

*Index b*and*Index a*, also derive from a neo-additive capacity and can therefore be linked to*α*and*δ*. That is, for their a-insensitivity measure:*Index*\( a \)=\( \delta \). For their ambiguity aversion measure:*Index b*= (2*α*− 1)*δ*. Hence, Index*b*is positive if and only if*α*> 0.5 and*δ*> 0.For now, we defer discussion of the ambiguity loss question (

*k*= 4) to Section 4.4.Out of the 3258 original respondents, 3 did not answer any questions, 85 did not complete all of our ambiguity questions, and 179 spent less than two minutes on answering the ambiguity questions. After excluding these 267 respondents, we have a final sample of 2991 respondents.

Pooled OLS estimates are consistent in the presence of random effects, but the standard errors may be inefficient. As we use clustered (robust) standard errors, the results of pooled OLS are similar to a random effects model.

Our estimate of

*α*is similar to values of*α*= 0.515 reported in Ahn et al. (2014) for a small sample of students and*α*= 0.556 in Potamites and Zhang (2012) for Chinese investors. Baillon et al. (2015) estimate*α*= 0.61 and*δ*= 0.51 in a sample of 64 students, with the source of ambiguity being the returns of an unknown stock.We test the single restriction

*δ*= 1, implied by all three models with [0,1] as the prior set. Joint tests of*δ*= 1,*α*= 1 for MaxMin-[0,1], or*δ*= 1,*α*= 0 for MaxMax-[0,1] give the same result.The equations for

*m*_{ ik }and*m*^{L}_{ ik }have different constant terms, (1–*α*)*δ*and*α*^{L}*δ*, so the model in Equation (11) is no longer applicable (it would imply the restriction 1*–α*=*α*^{L}). Introducing a dummy variable for the loss question permits us to separately estimate and identify*α*and*α*^{L}.A drawback of the model in Equation (12) is that the random effect

*u*_{ i }has opposite effects on ambiguity aversion for gains and losses, an assumption inconsistent with the positive correlation between*AA*^{50}and*AA*^{−50}. As a result the estimated correlation of the random effect (*ρ*) is relatively low in Table 3. We have also estimated a model with two separate random effects for the constant*c*and loss dummy*d*^{L}, but we find no difference in the main results concerning*α*,*α*^{L}and*δ*. Results are available on request.If we could measure more matching probabilities for ambiguous events involving loss outcomes with other likelihoods (e.g., similar to the 10 % and 90 % gains questions), we could also estimate

*δ*separately in the loss domain. We leave to future research additional refinements of ambiguity surveys and tests for reference dependence.Borghans et al. (2009) find that men are more ambiguity averse than women in a sample of 347 high school students. In a study of the Dutch population, Dimmock et al. (2015b) estimate the relation between ambiguity attitudes and control variables; there, however, few effects are statistically significant (sample size:

*N*= 666). Using our ALP Module, Dimmock et al. (2015a) show in a web appendix that the non-parametric ambiguity aversion measure*AA*^{50}is higher for men than for women, and positively related to risk aversion.This is measured on a reversed scale from 0 to 5, with higher values indicating

*lower*trust.As in Tanaka et al. (2010), utility is defined over the payoffs of the gambles (not integrated with total wealth), and the power coefficient is limited to the range from 0 to 1.5. Risk aversion, defined as ‘1 – power function coefficient’, varies from −0.5 (risk seeking) to +1 (strongest level of risk aversion), and a value of zero implies risk neutrality.

The derivative of

*α*_{ i }with of respect to*x*_{ ih }is : −*c*_{ h }/*δ*_{ i }−*s*_{ h }(*c*_{0}+ ∑^{H}_{ h = 1}*c*_{ h }*x*_{ ih })/*δ*^{2}_{ i }, which we evaluate at the mean values of*x*_{ ih }and δ_{ i }.

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## Acknowledgments

The survey module fielded by the authors in the RAND American Life Panel (ALP) was approved by the Institutional Review Board of the University of Pennsylvania. The authors gratefully acknowledge financial support from Netspar, and grants to the University of Pennsylvania from the National Institute on Aging (P30 AG-012836-18), and a grant from the National Institutes of Health–National Institute of Child Health and Development Population Research Infrastructure Program (R24 HD-044964-9). Support was also provided by the Pension Research Council/Boettner Center and the Wharton Behavioral Labs at the University of Pennsylvania. We also thank the ALP teams at RAND and the University of Southern California. We are grateful to Aurelien Baillon, Peter Wakker and participants at FUR 2014 for helpful comments, and to Tania Gutsche, Arie Kapteyn, Bart Orriens, and Bas Weerman for assistance with the survey. Yong Yu provided outstanding programming assistance. The content is solely the responsibility of the authors and does not represent the official views of the National Institute of Aging, the National Institutes of Health, or any of the other institutions providing funding for this study or with which the authors are affiliated.

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Dimmock, S.G., Kouwenberg, R., Mitchell, O.S. *et al.* Estimating ambiguity preferences and perceptions in multiple prior models: Evidence from the field.
*J Risk Uncertain* **51**, 219–244 (2015). https://doi.org/10.1007/s11166-015-9227-2

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DOI: https://doi.org/10.1007/s11166-015-9227-2